Download LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – PHYSICS
FOURTH SEMESTER – April 2009
XC 15
PH 4502 - MATHEMATICAL PHYSICS
Date & Time: 24/04/2009 / 9:00 - 12:00 Dept. No.
PART-A
Max. : 100 Marks
(10 x 2 = 20 MARKS)
ANSWER ALL QUESTIONS.
1). Given z1 = 2 – i and z2 = 2 + i find z1* z2 .
2). Check if the function f(z) = x + i y is analytic.
i 1
3). Evaluate
 zdz .
i
4). Explain the property of linearity in complex line integral.

5). Define the eigen value problem for the operator
.
x
6). Write down the two dimensional wave equation.
7). Give the Parseval’s identity for Fourier transforms.
8). Define Fourier sine transform.
9). Why is Lagrange’s interpolation advantageous over Newton’s interpolation?
10). Write down Simpson’s 1/3 rule for integration.
PART-B
( 4 x 7.5 = 30 MARKS).
ANSWER ANY FOUR QUESTIONS.
11). Simplify the following a). (4+ 2i) (2 + i) ; b). 4[(2+2i)/(2-2i)]2 – 3[(2-2i)/(2+2i)]2,
Locate these points in the complex plane.
12). Verify Cauchy’s integral theorem for the integral of z 2 over the boundary of the
rectangle with vertices (0,0) , (1,0) , (1,1), (0,1) in the counterclockwise sense.
13). Find D’Alembert’s solution of the wave equation for a vibrating string.
14). Prove the following for the Fourier transforms F{f(ax)}= (1/a)F(s/a) and F{f’(x)}= is
F(s), here F(s) is the Fourier transform of f(x) and the prime denotes differentiation
with respect to `x’.
dy
 y  x with y(0) = 2 Find y(0.2) with h = 0.1.
15). Use Euler method to solve
dx
PART-C
(4 x 12.5 = 50 MARKS)
ANSWER ANY FOUR QUESTIONS.
16). State and prove Cauchy’s integral formula.
17). Derive the Cauchy Riemann equation for a complex function to be analytic. Express
it in polar coordinates.
18). Explain the method of separation of variables to solve the one dimensional wave
 2u
 2u
equation 2  c 2 .
t
x
Check whether u = x2 – y2 satisfies the two dimensional Laplace equation.
19). (a). State and prove the convolution theorem for the Fourier transforms.
(b). Find the Fourier sine transform of e-ax.
20. (a). Given y = sin (x ) , generate the table for x = 0  /4 and  /2 Find the value of
sin (  /6). using Lagrange’s interpolation method.
(b). For the given data calculate the Newton’s forward difference table.
(x,y): (0,0), (1,2), (2,6), (3,16).
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